Global Well-Posedness and Scattering for the Defocusing Energy-Critical Wave Equation on Hyperbolic Spaces

We establish global well-posedness and scattering for the defocusing energy-critical nonlinear wave equation on n-dimensional real hyperbolic space Hn for n 3. The equation is formulated with the conformal Laplacian to preserve the scaling invariance of the energy functional. We overcome the failure of global Euclidean scaling symmetries by exploiting the spectral gap of the shifted Laplace-Beltrami operator and establishing a family of dispersive and Strichartz estimates intrinsic to hyperbolic geometry. By adapting the concentration-compactness rigidity argument of Kenig and Merle, we prove that any solution with finite critical energy exists globally and scatters to a linear solution as t . A key component of our analysis involves a profile decomposition in Hn that accounts for the non-compactness of the isometry group and the dispersive nature of the wave propagator on negatively curved manifolds.